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Question

# If f(x)={|x|−3,x<1|x−2|+a,x≥1 and g(x)={2−|x|,x<2sgn(x)−b,x≥2, where sgn(x) denotes the signum function. If h(x)=f(x)+g(x) is discontinuous at exactly one point, then which of the following values of a and b are possible?

A
a=3,b=0
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B
a=2,b=1
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C
a=2,b=0
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D
a=3,b=1
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Solution

## The correct options are A a=−3,b=0 B a=2,b=1f(x)={|x|−3,x<1|x−2|+a,x≥1 f(x) is continuous for all x if it is continuous at x=1, for which limx→1−f(x)=limx→1+f(x)=f(1) ⇒|1|−3=|1−2|+a ⇒a=−3 g(x)={2−|x|,x<2sgn(x)−b,x≥2 g(x) is continuous for all x if it is continuous at x=2, for which limx→2−g(x)=limx→2+g(x)=g(2) ⇒2−|2|=sgn(2)−b=1−b ⇒b=1 Thus, h(x)=f(x)+g(x) is continuous for all x if a=−3, b=1 Hence, h(x) is discontinuous at exactly one point for (a=−3,b=0) and (a=2,b=1)

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